Finite Group Databasebeta

← #34199162

Subgroups — #34199162

Order 20224.

382 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23763477C211yesmaximal, characteristic
2#23761714C211yesmaximal, characteristic
3#23761424C212yesmaximal
4#23761400C212yesmaximal
5C79×#2611C211yesmaximal, characteristic
6#19098616—21no—
7#19098426—21no—
8#19098330—21no—
9#19098330C2212yes—
10#19098289—21no—
11#19098287C2212yes—
12C2×#125×C79C411yescharacteristic, complemented
13C2×#117×C79C2211yescharacteristic
14#400×C79C411yescharacteristic, complemented
15#399×C79C2211yescharacteristic
16#398×C79C411yescharacteristic, complemented
17#397×C79C2211yescharacteristic
18#389×C79—21no—
19#388×C79—21no—
20#387×C79—21no—
21#385×C79—21no—
22#376×C79C411yescharacteristic, complemented
23C2×C22⋊Dic79—22no—
24C4⋊(C79⋊C8)—41no—
25C22⋊(C79⋊C8)—22no—
26C4×C79⋊C8—41no—
27C79⋊#95—41no—
28C79⋊#95—41no—
29C23×C4×C79D411yescharacteristic
30#125×C79—41no—
31#125×C79—41no—
32#117×C79—41no—
33#117×C79—41no—
34C2×C4⋊C4×C79—21no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79C2×C411yescharacteristic
39C2×C4⋊C4×C79C2×C411yescharacteristic
40C2×C4⋊C4×C79C2×C411yescharacteristic
41C2×C4⋊C4×C79D411yescharacteristic
42C2×C22⋊C4×C79—21no—
43C2×C22⋊C4×C79—21no—
44C2×C22⋊C4×C79D411yescharacteristic
45C2×C22⋊C4×C79C2311yescharacteristic
46C2×C42×C79—21no—
47C2×C42×C79C2×C411yescharacteristic
48C2×C42×C79D411yescharacteristic
49#95×C79—21no—
50#95×C79—21no—
51#95×C79—21no—
52#95×C79—21no—
53#95×C79—21no—
54#95×C79C2×C411yescharacteristic
55#95×C79C2×C411yescharacteristic
56C22×Dic79—42no—
57C2×C79⋊C8—42no—
58C22⋊Dic79—42no—
59C22⋊Dic79—42no—
60C24×C79C2×D411yescharacteristic
61C22×C4×C79—21no—
62C22×C4×C79—22no—
63C22×C4×C79—21no—
64C22×C4×C79—21no—
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—41no—
69C22×C4×C79C22⋊C411yescharacteristic
70C22×C4×C79C22⋊C411yescharacteristic
71C22×C4×C79C22⋊C411yescharacteristic
72C22×C4×C79C22⋊C411yescharacteristic
73C22×C4×C79C22×C411yescharacteristic
74C22×C4×C79C2×D411yescharacteristic
75C4⋊C4×C79—41no—
76C4⋊C4×C79—41no—
77C4⋊C4×C79—42no—
78C4⋊C4×C79—42no—
79C4⋊C4×C79—42no—
80C4⋊C4×C79—41no—
81C4⋊C4×C79—41no—
82C4⋊C4×C79—41no—
83C4⋊C4×C79—41no—
84C22⋊C4×C79—41no—
85C22⋊C4×C79—41no—
86C22⋊C4×C79—42no—
87C22⋊C4×C79—42no—
88C42×C79—21no—
89C42×C79—21no—
90C42×C79—21no—
91C42×C79—21no—
92C42×C79—42no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79C2×C22⋊C411yescharacteristic
98C23×C79#13512yes—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—42no—
110C2×C4×C79—42no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79#9911yescharacteristic
127C2×C4×C79#9911yescharacteristic
128C2×Dic79—82no—
129C2×Dic79—82no—
130C2×Dic79—82no—
131C79⋊C8—82no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—22no—
135C22×C79—42no—
136C22×C79—41no—
137C22×C79—41no—
138C22×C79—81no—
139C22×C79C2×#9911yescharacteristic
140C22×C79#41611yescharacteristic
141C22×C79#42111yescharacteristic
142C4×C79—21no—
143C4×C79—21no—
144C4×C79—42no—
145C4×C79—41no—
146C4×C79—41no—
147C4×C79—41no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79—81no—
151C4×C79—81no—
152C4×C79—81no—
153Dic79—162no—
154#11846—791nomaximal, Sylow
155C2×C79—21no—
156C2×C79—21no—
157C2×C79—42no—
158C2×C79#151611yescharacteristic
159C2×C79#212311yescharacteristic
160C2×C79#212611yescharacteristic
161#2611D7911yescharacteristic
162#1820—791no—
163#1458—791no—
164#1352—792no—
165#1344—792no—
166C79#1184611yescharacteristic, complemented, Sylow
167C2×#125Dic7911yescharacteristic, complemented
168C2×#117C2×D7911yescharacteristic
169#431—1581no—
170#400Dic7911yescharacteristic, complemented
171#399C2×D7911yescharacteristic
172#398Dic7911yescharacteristic, complemented
173#397C2×D7911yescharacteristic
174#389—21no—
175#388—21no—
176#387—21no—
177#385—21no—
178#381—1581no—
179#376Dic7911yescharacteristic, complemented
180#342—792no—
181#342—1581no—
182#324—1581no—
183#323—792no—
184C23×C4C22⋊D7911yescharacteristic
185#125—41no—
186#125—41no—
187#117—41no—
188#117—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic7911yescharacteristic
194C2×C4⋊C4C2×Dic7911yescharacteristic
195C2×C4⋊C4C2×Dic7911yescharacteristic
196C2×C4⋊C4C22⋊D7911yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—1582no—
200C2×C22⋊C4C22⋊D7911yescharacteristic
201C2×C22⋊C4C22×D7911yescharacteristic
202C2×C42—21no—
203C2×C42C2×Dic7911yescharacteristic
204C2×C42C22⋊D7911yescharacteristic
205C4⋊C8—3161no—
206C22⋊C8—1582no—
207C4×C8—3161no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—3161no—
214#95—3161no—
215#95C2×Dic7911yescharacteristic
216#95C2×Dic7911yescharacteristic
217C24C2×C22⋊D7911yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—3162no—
227C22×C4C22⋊Dic7911yescharacteristic
228C22×C4C22⋊Dic7911yescharacteristic
229C22×C4C22⋊Dic7911yescharacteristic
230C22×C4C22⋊Dic7911yescharacteristic
231C22×C4C22×Dic7911yescharacteristic
232C22×C4C2×C22⋊D7911yescharacteristic
233C2×C8—3162no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—3162no—
248C22⋊C4—3162no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—42no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊Dic7911yescharacteristic
259C23#1457918112yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—6322no—
288C2×C4—6322no—
289C2×C4—6322no—
290C2×C4#1457907611yescharacteristic
291C2×C4#1457907611yescharacteristic
292C8—6322no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#1457907611yescharacteristic
301C22#1909855911yescharacteristic
302C22#1909858011yescharacteristic
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—12642no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2376195611yescharacteristic
319C2#2376505411yescharacteristic
320C2#2376506511yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2611C2
socleC22×C79C2×#99
prime radicalC24×C79C2×D4
cosocleC2×C22⋊C4×C79C23
hypercenter#2611D79
hypocenterC79#11846
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. C2×#125×C79C4semidirect
14. #400×C79C4semidirect
16. #398×C79C4semidirect
22. #376×C79C4semidirect
166. C79#11846semidirect
167. C2×#125Dic79semidirect
170. #400Dic79semidirect
172. #398Dic79semidirect
179. #376Dic79semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237634772
#237617142
#237614242
#237614002
C79×#26112
#1184679